The Isoperimetric Inequality On The Sphere
Let C be a simple closed curve on a sphere of radius 1. Denote by L the length of C and by A the area enclosed by C. The spherical isoperimetric inequality states that
and that the equality holds if and only if the curve is a circle. There are, in fact, two ways to measure the spherical area enclosed by a simple closed curve, but the inequality is symmetric with the respect to taking the complement.
This inequality was discovered by Paul Lévy (1919) who also extended it to higher dimensions and general surfaces.
Read more about this topic: Isoperimetric Inequality
Famous quotes containing the words inequality and/or sphere:
“All the aspects of this desert are beautiful, whether you behold it in fair weather or foul, or when the sun is just breaking out after a storm, and shining on its moist surface in the distance, it is so white, and pure, and level, and each slight inequality and track is so distinctly revealed; and when your eyes slide off this, they fall on the ocean.”
—Henry David Thoreau (1817–1862)
“A knowledge of the Globe and its various inhabitants, however slight ... has a kindred effect with that of seeing them as travellers, which never fails, in uncorrupted minds, to weaken local prejudices, and enlarge the sphere of benevolent feelings.”
—James Madison (1751–1836)